Find the characteristic polynomial of an arbitrary-sized matrix

Question: Let be the matrix defined by for all . Find the characteristic polynomial of .

My attempt: Call to be the matrix of all 's. Then its characteristic polynomial . Use proof by induction.

Base case: . Then its characteristic polynomial . This checks.

Then for induction step, assume that it holds for and attempt to prove it to be true for . My problem with this was that as I expanded out the determinant along the top row of , things got very messy and I couldn't see a pattern.

Could you give me a hand, please? Thanks!